Ultrawideband Dispersion Control of a Metamaterial
Surface for Perfectly-Matched-Layer-Like Absorption
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Ye, Dexin, Zhiyu Wang, Kuiwen Xu, Huan Li, Jiangtao Huangfu,
Zheng Wang, and Lixin Ran. “Ultrawideband Dispersion Control
of a Metamaterial Surface for Perfectly-Matched-Layer-Like
Absorption.” Physical Review Letters 111, no. 18 (October 2013).
© 2013 American Physical Society
As Published
http://dx.doi.org/10.1103/PhysRevLett.111.187402
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PRL 111, 187402 (2013)
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PHYSICAL REVIEW LETTERS
Ultrawideband Dispersion Control of a Metamaterial Surface
for Perfectly-Matched-Layer-Like Absorption
Dexin Ye,1,2 Zhiyu Wang,1,2 Kuiwen Xu,1 Huan Li,1 Jiangtao Huangfu,1 Zheng Wang,3 and Lixin Ran1,*
1
Laboratory of Applied Research on Electromagnetics (ARE), Zhejiang University, Hangzhou 310027, China
2
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
3
Department of Electrical and Computer Engineering, Microelectronics Research Center, the University of Texas at Austin,
Texas 78758, USA
(Received 13 January 2013; revised manuscript received 5 August 2013; published 29 October 2013)
Narrow bandwidth is a fundamental issue plaguing practical applications of metamaterial absorbers. In
this Letter, we show that by deliberately controlling the dispersion and dissipation of a metamaterial, an
ultrawideband perfect metamaterial absorber with complex-valued constitutive parameters strictly satisfying the modified model of a perfectly matched layer, can be achieved. The nearly perfect power
absorption, better than 99%, was experimentally observed in an unprecedented bandwidth of 39%,
approaching the theoretical Rozanov limit. We expect a wide range of applications to emerge from this
general concept.
DOI: 10.1103/PhysRevLett.111.187402
PACS numbers: 78.68.+m, 41.20.q, 78.20.Ci
Light propagation, focusing, and scattering are governed
by the electromagnetic properties of the media being traversed [1]. These properties, either occurring naturally or
synthesized with artificial metamaterials [2], must obey
causality, which manifests as the Kramers-Kronig (K-K)
relations between the real and imaginary parts of the
frequency-dependent constitutive parameters, such as permittivity and permeability [3]. In natural media, dispersion
and dissipation are determined by the underlying electronic
and molecular energy levels, with largely limited tunability. In contrast, metamaterials are made of densely
arranged subwavelength resonance cells, each scalable
over decades of frequencies throughout microwave and
optical regimes. The associated permittivity and permeability, either positive or negative, can be freely tuned over
the entire range [4–6]. In the past decade, tremendous
interest has risen in applying these unique dispersions
toward novel applications, such as superlenses [7], invisibility cloaks [8], and perfect absorbers [9–18]. However,
since these attractive applications rely on strong dispersions, they suffer from narrow bandwidth operation, which
is a fundamental obstacle constraining metamaterials in
practical applications.
In this Letter, we point out that a metamaterial can be
created to have frequency-independent properties over an
ultrawide frequency range, while maintaining its ability to
exhibit electromagnetic characteristics that cannot be found
in natural media. Based on the K-K relations, this can be
achieved by combining multiple high-Q and low-Q resonances with deliberately controlled losses. To validate this
approach, we experimentally demonstrated a metamaterial
perfect absorber (MPA), whose frequency-independent dispersion regions of both permittivity and permeability are
stretched to an ultrawide band, in which both the real and
imaginary parts of the complex constitutive parameters are
0031-9007=13=111(18)=187402(5)
precisely tuned to satisfy a modified model of a perfectly
matched layer (PML). Experimental data show that with a
thickness of 0.2 free-space wavelength, such an MPA can
achieve an unprecedented relative bandwidth of 39%, over
which the power absorption remains over 99%. Analysis
shows that the thickness of the MPA approaches the theoretical Rozanov limit, which provides the lower bound of
the thickness of any physical absorber [19].
It is well-known that metamaterials exhibit Lorentzian
dispersions, whose constitutive parameter ð!Þ, either the
permittivity "ð!Þ or the permeability ð!Þ, obeys the K-K
relations [3–5]:
Reðð!ÞÞ ¼ 1 þ 2=
Z þ1
0
d!0 Imðð!0 ÞÞ!0 =ð!02 !2 Þ;
(1.1)
Imðð!ÞÞ ¼ 2!=
Z þ1
0
d!0 ½Reðð!0 ÞÞ 1=ð!02 !2 Þ:
(1.2)
The K-K relations imply that the real part of a
constitutive parameter can be tuned by controlling the
imaginary part (or vice versa). Specifically, a metamaterial
consisting of engineered subwavelength-scale cells makes
it possible to modify dispersion by deliberately introducing
multiple resonances with controlled dissipation. In what
follows, we compare two analytical examples to illustrate
how ultrawideband flat dispersions of constitutive parameters, where the real parts are either less than unity or
negative, can be obtained by tuning three adjacent resonances. Subsequently, we show by full-wave simulations
and experiments that an ultrawideband MPA with
equal complex-valued constitutive parameters can be
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FIG. 1 (color online). Dispersion engineering of Lorentzian resonances to achieve a nearly constant positive value less than unity (a)
and a negative value (b) within a wide target frequency range, where panels I, II, and III show the process of the dispersion control by
varying the damping frequency.
implemented by introducing five electric and magnetic
resonances in each unit cell.
A Lorentz model with three resonances can be expressed
by
ð!Þ ¼ 1 !21p =ð!2 !21 þ i!1 Þ
!22p =ð!2 !22 þ i!2 Þ
!23p =ð!2 !23 þ i!3 Þ;
(2)
where !i , !ip and, i , are the resonance, plasma, and
damping frequencies of the i-th resonance region (i ¼ 1,
2, 3), respectively. Starting from !1 ¼ !0 , !2 ¼ 2!0 ,
!3 ¼ 3!0 , !1p ¼ !2p ¼ !3p ¼ 3!0 , and an initial set
of damping frequencies of 1 ¼ 0:25!0 , 2 ¼ 0:15!0 ,
and 3 ¼ 0:1!0 , ð!Þ, three resonance regions can be
seen in panel I of Fig. 1(a). With small damping frequencies, the absorption exhibits a sharp dispersion spectrum
near the resonance. In panel II, we see that after increasing
the damping frequencies to 1 ¼ 0:3!0 , 2 ¼ 0:8!0 , and
3 ¼ 0:1!0 , anomalous regions are spread out separately
and the second dispersion region becomes flatter.
Increasing the damping frequencies further to 1 ¼
0:85!0 , 2 ¼ 1:7!0 , 3 ¼ 0:5!0 , the real part of ð!Þ
is controlled to maintain a nearly constant positive value
less than unity between !1 and !3 , as shown in panel III.
In the same way, we also control the damping frequencies
to achieve a nearly constant negative real part of ð!Þ, as
shown in Fig. 1(b). Assume !1 ¼ !0 , !2 ¼ 1:48!0 , !3 ¼
2:2!0 , and !1p ¼ !2p ¼ !3p ¼ 2!0 , when 1 , 2 , and
3 are separately increased as indicated in panels I, II, and
III, we can finally obtain a nearly constant, negative real
part of ð!Þ. Note that in the shaded regions in Figs. 1(a)
and 1(b), both relative bandwidths exceed 40%.
Recently, researches on new concept absorbers
have attracted much attention [20–23]. Among them,
MPAs show potential advantages in achieving nearly perfect absorption with subwavelength thickness [9–18].
Particularly, when an MPA’s constitutive parameters are
tuned to strictly satisfy a modified uniaxial PML model,
the MPA maintains a nearly perfect absorption for normal
and small-angle incidences [14]. Nevertheless, similar to
other existing metamaterial applications, MPAs also suffer
from limited bandwidths. To absorb energy in a thin layer
without incurring any reflection, an MPA has to have large
and equal imaginary parts of the permittivity and permeability, necessitating strong dispersions within the same
band. Although engineering approaches can be used to
combine multiple narrow band resonances [15–18], the
overall absorption performances have not yet been comparable to PMLs. However, by making use of the dispersion
control, we can realize a single-layer MPA with nearly
perfect ultrawideband absorption. Such PML-like
responses were previously only seen in theoretical studies
with mathematically defined dispersions.
In order to obtain a perfect absorption over an ultrawide
band, the complex permittivity and permeability tensors
need to be identical over the specified band [14]. The
identical tensors have a diagonal form diagð; ; hÞ, where
h represents the permittivity and permeability component
along the optical axis. To accomplish a nearly perfect
absorption over a subwavelength material thickness, a
large, positive imaginary part 00 for the complex-valued
( ¼ 0 þ i00 ) is required (See Supplemental Material
[24]). For this purpose, we begin with the basic subwavelength unit cell consisting of metallic split-ring resonator
(SRR) and rod, shown as ‘‘Cell 1’’ in Fig. 2(a) [25]. To
introduce multiple electric and magnetic resonances without increasing the size of the cell, we split the original
SRRs into four smaller, centrosymmetric SRRs, as illustrated by ‘‘Cell 2’’. To introduce independently tunable
loss for each SRR, the outside corner of each SRR is cut
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FIG. 2 (color online). Design of the unit cell of an ultrawideband MPA. (a) Introducing multiple resonances in an SRR and
rod cell. (b) Topology of the new cell and related equivalent
circuit model. (c) Retrieved constitutive parameters for different
resistances.
and a lumped resistor is soldered, as shown by ‘‘Cell 3’’.
Meanwhile, the rod is extended in the transverse plane to
introduce a stronger electric resonance and provide a
ground plane required by the uniaxial PML model [26,27].
The final topology of the new cell is shown in Fig. 2(b).
For a y-polarized magnetic incidence, the equivalent circuit model for the top-left SRR is shown in the inset, where
R1 is the resistance introduced to tune the resonance, C1
and C2 are the capacitances of the gaps at the upper-left
and lower-right corners, and L is the inductance determined by the length and width of the strips between the
two gaps. By tuning the length, width of the strips, and
especially, the intentionally introduced lumped resistance,
one can tune both electric and magnetic resonances, and
consequently, tune the real and imaginary parts of both
permittivity and permeability tensors of the metamaterial.
To clearly show the effect of the dispersion control, we
start by simulating an MPA consisting of the cells shown in
Fig. 2(b). In each cell, four SRRs are printed on a 1.5 mm
thick FR4 substrate with a dielectric constant of 4.4 and a
loss tangent of 0.02. To ensure commercial surface-mount
resistors exhibit negligible reactance, the dimensions of the
unit cell are chosen to be a ¼ 40 mm, b ¼ 1:2 mm, c ¼
0:4 mm, d ¼ 16:4 mm, and e ¼ 8 mm, such that multiple
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resonances appear in a band centered at 1.5 GHz. Around
this frequency, the periodicity of the unit cells is =5, and
the MPA can be treated as an effective medium to retrieve
effective constitutive parameters using a homogenization
algorithm [28,29]. To retrieve the dispersions of the complex constitutive parameters from simulation data, we use a
commercial Maxwell-equations solver, CST MWS, to perform full-wave simulations. Details of the parameter
retrieval are provided in Supplemental Material [24].
In the process of controlling the dispersion, we first set
the values of all four resistors to 1 Ohm, to obtain initial
frequencies of the multiple resonance. In the panel I in
Fig. 2(c), we find three magnetic resonances and two
electric resonances in the band from 500 MHz to
2.5 GHz. Note that each resonance has a narrow bandwidth. After increasing the four resistances to 10 Ohm, we
find in panel II that all the bandwidths of the resonances are
broadened, in agreement with our theoretical analysis.
When the four resistances are increased to 100 Ohm, we
find in panel III that the second magnetic resonance and the
two electric resonances inside the grey region become
indistinguishable due to the increased dissipation, and
both the imaginary and real parts inside the grey region
become flatter. Finally, to achieve nearly perfect absorption
in this band, the dispersions of both complex permittivity
"r and permeability r have to obey the modified PML
model in the whole band, i.e., Re½"r ¼ Re½r , Im½"r ¼
Im½r , and Im½"r and Im½r should be larger than 2
(See Supplemental Material [24]). By performing a parameter sweep of the four resistances, we obtain an optimal
set of 203, 122, 39, and 63 Ohms for resistors 1 to 4,
respectively. With these values, the dispersions of the
complex permittivity and permeability are nearly equal in
the frequency band from 1.2 to 1.8 GHz, as shown in panel
IV. In this band, the real parts of permittivity and permeability approach zero, while the imaginary parts are all
larger than 5. Comparing with the absorption dependence
with the thickness and imaginary parts shown in Fig. S1 in
Supplemental Material [24], we achieve a nearly perfect
ultrawideband absorption.
Based on the full-wave simulation, we show the effective wave impedance of the MPA layer and the greatly
suppressed reflection coefficient jS11 j at normal incidence
[Figs. 3(a) and 3(b)]. The near-unity wave impedance of
the MPA layer appears over the entire grey region (from
1.25 to 1.84 GHz), nearly the same as the impedance of air.
The consequence is the lower than 30 dB reflection
coefficient in the same region. Figure 3(c) plots the total
power absorption inside the MPA layer and the power
consumed by the resistors and other materials (i.e., substrate and the copper strips). We see that in the simulation,
almost all the incident power is consumed by resistors,
and the total power absorption is higher than 99.9% in a
0.59-GHz band from 1.25 to 1.84 GHz, corresponding to a
relative bandwidth about 38% with respect to the central
frequency. For the power absorption higher than 99%,
the relative bandwidth increases to 50%. The lumped
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FIG. 3 (color online). Simulated wave impedance (a), reflection coefficient (b), and frequency-dependent power absorption
for different resistors (c).
resistances play a key role in the precise dispersion control
to achieve nearly perfect absorption.
To fabricate a polarization-independent MPA sample,
the FR4 boards printed with the SRR units are cut and
assembled into an interlocking lattice. The implemented
520 520 mm2 -sized sample includes 312 unit cells and
1248 surface-mount resistors, as shown in the inset of
Fig. 4. Due to the cross coupling induced by the interlock,
resistors R1 , R2 , R3 , and R4 have been tuned to be 226, 324,
280, and 63.5 Ohms, respectively. The simulated absorption of the new MPA layer is given in Fig. 4. With these
modified resistances, the 99.9% bandwidth of the power
absorption is now 0.47 GHz (from 1.27 to 1.75 GHz),
corresponding to a relative bandwidth of 31%.
We measure the power reflection and absorption of the
MPA sample in a microwave anechoic chamber using an
Agilent’s 8722ES network analyzer and a pair of HD
microwave’s wideband horn antennas. For normal incidence, the reflection is measured with the MPA sample
placed 1.5 meters away from the antenna, and the result is
FIG. 4 (color online). Full-wave simulation of the fabricated
MPA sample.
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normalized with a control experiment using a copper board
of an identical size. The center of the sample is carefully
aligned with the center of the antenna aperture, as shown in
the inset of Fig. 5(a). The normalized power absorption
for both y- and x-polarized incidences are calculated by
1-jS11 j2 , and are compared with simulation results in
Fig. 5(a). We see that for the x-polarized incidence (blue
line), the bandwidth in which the absorption is larger than
99% is 0.49 GHz (from 1.3 GHz to 1.79 GHz), corresponding to a relative bandwidth of 31.5%. For a y-polarized
incidence (red line), the power absorption is similar to the x
polarization.
Reflection at oblique incidence was then measured using
two horn antennas as transmitter and receiver. The antennas were separated by an absorber screen to reduce mutual
coupling, as shown in the inset of Fig. 5(a). The reflection
from the MPA sample, as well as the copper board previously used in the control experiment, are separately
measured as the incident angles change from 0 to 45.
Figure 5(b) plots the normalized power reflection versus
incident angles. We see that in the cases when incident
angles are smaller than 30 , the power absorption below
20 dB still occurs over an ultrawide bandwidth of
0.59 GHz (from 1.22 to 1.81 GHz), corresponding to a
relative bandwidth of 39%.
For a physically realizable broadband nonmagnetic
absorber, the Rozanov limit indicates that for any metal
backed absorber under a normal incidence, its total thickness d must be larger than a theoretical limit for the given
frequency response of the absorption [19], i.e.,
Z 1
lnjRðÞjd
(3)
22 d;
0
where R is the reflection coefficient and is the free-space
wavelength. Note that our absorber consists of
FIG. 5 (color online). Experimental absorption of the fabricated MPA sample. (a) Normal incidence. (b) Oblique
incidences.
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nonmagnetic copper wires and dielectric substrates, and
therefore applies to Rozanov analysis. For the broadband
absorption shown in Fig. 3(c), the corresponding theoretical limit is estimated to be 2.8 cm (given the data available,
the integration is performed from 0.5 to 2.5 GHz). This
value, likely underestimated, is close to the thickness
(4 cm) of the designed MPA layer. For the measured data
shown in Fig. 5(a), the calculated lower bound is about
2 cm, with the integration from 0.8 to 2.2 GHz. Although
the MPA sample is slightly thicker than the underestimated
lower bound, the =5-thick MPA sample is much thinner
than pyramidal absorbers, which are typically thicker than
20 cm for the same operating band [30].
In conclusion, we show that the dispersions of the complex constitutive parameters of a metamaterial can be
deliberately controlled to produce flat, ultrawide band
frequency responses. In the example of implementing an
ultrawideband MPA, we experimentally demonstrated
highly precise dispersion control, capable of approximating a modified PML model requiring a strict, ultrawideband complex equality of the permittivity and permeability
tensors. No such MPAs have been previously reported to
accomplish this strict condition, nor has PML-like absorption been observed. It is worth noting that this approach
also applies to achieving ultrawideband dissipation by
controlling the real parts with tunable lumped reactance,
to potentially obtain imaginary parts with novel frequency
responses, such as mimicking a graphenelike conductivity
in specified frequency ranges [31]. Based on the recent
advancement of optical lumped nanocircuits [32,33], our
approach may be extended to infrared and optical bands.
We also expect several applications of this general principle of artificial dispersion control in the realization of
various ultrawideband metamaterials.
This work is sponsored by the NSFC under Grants
No. 61131002 and No. 61071063. Z. W. acknowledges
support from the Alfred P. Sloan Research Fellowship.
*Corresponding author.
ranlx@zju.edu.cn
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